2.1 Governing equations and boundary conditions

Under the paradigm of the leaky dielectric model, the bulk fluid is charge free (Saville Reference Saville1997) and the electrostatic problem is governed by the Laplace equation for the electric potential both inside and outside the drop in the following form

(2.2a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D711}_{i}=0,\quad \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D711}_{e}=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D711}_{i}$ and $\unicode[STIX]{x1D711}_{i}$ represent the electric potential inside and outside the drop, respectively. The electric potential outside the drop ( $\unicode[STIX]{x1D711}_{e}$ ) is bounded and satisfies the following far-field condition

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{e}=-\boldsymbol{E}_{\infty }\boldsymbol{\cdot }\boldsymbol{x}=-r\cos \unicode[STIX]{x1D703}\quad \text{as}~r\rightarrow \infty .\end{eqnarray}$$

The electric potential inside the drop ( $\unicode[STIX]{x1D711}_{i}$ ) is bounded at the drop centre ( $r=0$ ). At the drop interface, the electric potential is continuous

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{i}=\unicode[STIX]{x1D711}_{e}\quad \text{at }r=r_{s}(\unicode[STIX]{x1D703}).\end{eqnarray}$$

Additionally, the electric potentials should satisfy the balance between ohmic conduction and charge convection at the drop interface (Mandal, Bandopadhyay & Chakraborty Reference Mandal, Bandopadhyay and Chakraborty2016b )

(2.5) $$\begin{eqnarray}\boldsymbol{n}\boldsymbol{\cdot }(R\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}_{i}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}_{e})=-Re_{E}\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }(q_{s}\boldsymbol{V}_{s})\quad \text{at }r=r_{s}(\unicode[STIX]{x1D703}),\end{eqnarray}$$

where $q_{s}=\boldsymbol{n}\boldsymbol{\cdot }(S\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}_{i}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}_{e})|_{r=r_{s}}$ represents the surface charge distribution and $\boldsymbol{V}_{s}$ is the surface velocity. In (2.5), $\boldsymbol{n}=\unicode[STIX]{x1D735}(r-r_{s})/|\unicode[STIX]{x1D735}(r-r_{s})|$ represents the outward unit normal to the drop interface and $\unicode[STIX]{x1D735}_{s}=\unicode[STIX]{x1D735}-\boldsymbol{n}(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735})$ represents the surface divergence operator. Note that if we substitute $R=S$ in the governing equations and the boundary conditions of the leaky dielectric system then the system is equivalent to a perfectly dielectric one (refer to appendix A for a detailed justification).

In the creeping flow limit (i.e. $Re=0$ ), the flow problem is governed by the Stokes equation and the condition of incompressibility in the following form

(2.6) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D735}p_{i}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}_{i},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i}=0,\\ \unicode[STIX]{x1D735}p_{e}=\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}_{e},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{e}=0,\end{array}\right\}\end{eqnarray}$$

where $p$ represents the pressure field and $\boldsymbol{u}=(u_{r},u_{\unicode[STIX]{x1D703}})$ represents the velocity field. Exploiting the symmetry of electric and flow problems about the z-axis, it is convenient to use the streamfunction $\unicode[STIX]{x1D6F9}$ which will simplify the mathematical calculations (Mandal, Bandopadhyay & Chakraborty Reference Mandal, Bandopadhyay and Chakraborty2017a ). The velocity components are related to the streamfunction as (Leal Reference Leal2007): $u_{r}=-(1/r^{2})\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}$ and $u_{\unicode[STIX]{x1D703}}=-(1/r\sqrt{1-\unicode[STIX]{x1D702}^{2}})\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}/\unicode[STIX]{x2202}r$ , where $\unicode[STIX]{x1D702}=\cos \unicode[STIX]{x1D703}$ is the transformed polar coordinate. The Stokes equations can be expressed in terms of the streamfunction (which automatically satisfies the incompressibility condition) in the following form

(2.7a,b ) $$\begin{eqnarray}{\mathcal{L}}^{2}({\mathcal{L}}^{2}\unicode[STIX]{x1D6F9}_{i})=0,\quad {\mathcal{L}}^{2}({\mathcal{L}}^{2}\unicode[STIX]{x1D6F9}_{e})=0,\end{eqnarray}$$

where ${\mathcal{L}}^{2}$ is a second-order linear operator of the form ${\mathcal{L}}^{2}=\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}r^{2}+\{(1-\unicode[STIX]{x1D702}^{2})/r^{2}\}\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}$ . The velocity field outside the drop ( $\boldsymbol{u}_{\boldsymbol{e}}$ ) is bounded and satisfies the far-field condition

(2.8) $$\begin{eqnarray}\boldsymbol{u}_{e}=\unicode[STIX]{x1D71E}\boldsymbol{\cdot }\boldsymbol{x}\quad \text{as }r\rightarrow \infty .\end{eqnarray}$$

The velocity field inside the drop ( $\boldsymbol{u}_{i}$ ) is bounded at the drop centre ( $r=0$ ). At the drop interface, the velocity field is continuous and the normal component of the velocity field vanishes

(2.9) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\boldsymbol{u}_{i}=\boldsymbol{u}_{e}\quad \text{at }r=r_{s}(\unicode[STIX]{x1D703}),\\ \boldsymbol{u}_{i}\boldsymbol{\cdot }\boldsymbol{n}=\boldsymbol{u}_{e}\boldsymbol{\cdot }\boldsymbol{n}=0\quad \text{at }r=r_{s}(\unicode[STIX]{x1D703}).\end{array}\right\}\end{eqnarray}$$

Additionally, the velocity and pressure fields should satisfy the balance of hydrodynamic, electric and capillary stresses at the drop interface

(2.10) $$\begin{eqnarray}(\unicode[STIX]{x1D749}_{e}^{H}+M\unicode[STIX]{x1D749}_{e}^{E})\boldsymbol{\cdot }\boldsymbol{n}-(\unicode[STIX]{x1D749}_{i}^{H}+M\unicode[STIX]{x1D749}_{i}^{E})\boldsymbol{\cdot }\boldsymbol{n}=\frac{1}{Ca}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{n}\quad \text{at }r=r_{s}(\unicode[STIX]{x1D703}),\end{eqnarray}$$

where $\unicode[STIX]{x1D749}^{H}$ and $\unicode[STIX]{x1D749}^{E}$ are the hydrodynamic and electric stress tensors (Mandal, Chakrabarti & Chakraborty Reference Mandal, Chakrabarti and Chakraborty2017b ).

2.2 Expansion in perturbation of $Re_{E}$ and $Ca$

A closer look into the electrostatic and flow problems reveals that both the electrostatic and flow problems are coupled and nonlinear in nature even though the governing equations are linear. The coupling between the electrostatic and flow problems stems from the nonlinear surface charge convection term (right-hand side of (2.5)). Another source of nonlinearity is present in all the interfacial boundary conditions due to the unknown shape of the drop (i.e. unknown $f$ ( $\unicode[STIX]{x1D703}$ )). Thus the EHD of the drop in the combined presence of a uniform electric field and uniaxial extensional flow is not a mere linear superposition of the EHD of a drop in a uniform electric field and the hydrodynamics of drop in uniaxial extensional flow. Due to the two-way coupled and nonlinear nature of the problem, an analytical solution cannot, in general, be obtained for arbitrary values of $Re_{E}$ , Ca and $M$ . To proceed with an analytical treatment, in the present study we focus on the particular case in which $1\gg Re_{E}$ , $Ca\gg Re_{E}^{2}$ , $Re_{E}Ca$ , $Ca^{2}$ and $M\sim 1$ (Bandopadhyay et al. Reference Bandopadhyay, Mandal, Kishore and Chakraborty2016). Physically it describes a system where the deformation of the drop is not far from a spherical shape and where the effect of the convection of the surface charges is weak as compared to the ohmic conduction. $M\sim 1$ represents the physical situation where the viscous stress and the electrical stress are of a comparable order of magnitude. These considerations allow us to represent any field variable $\unicode[STIX]{x1D712}$ in terms of a double asymptotic expansion using $Re_{E}$ and Ca as perturbation parameters in the following form:

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}^{(0)}+Re_{E}\unicode[STIX]{x1D712}^{(Re_{E})}+Ca\unicode[STIX]{x1D712}^{(Ca)}+\cdots \,,\end{eqnarray}$$

where $\unicode[STIX]{x1D712}^{(0)}$ is the leading-order contribution, $\unicode[STIX]{x1D712}^{(Re_{E})}$ is the $O(Re_{E})$ contribution and $\unicode[STIX]{x1D712}^{(Ca)}$ is the $O(Ca)$ contribution. Substituting the expansion given in (2.11) for all dependent variables into (2.2)–(2.10), one can easily obtain governing equations and respective boundary conditions at different orders of perturbation (Mandal, Bandopadhyay & Chakraborty Reference Mandal, Bandopadhyay and Chakraborty2016a ).

The unknown shape of the drop $r_{s}$ is also expanded as (Bandopadhyay et al. Reference Bandopadhyay, Mandal, Kishore and Chakraborty2016)

(2.12) $$\begin{eqnarray}r_{s}=1+f(\unicode[STIX]{x1D703})=1+Caf^{(Ca)}+CaRe_{E}f^{(CaRe_{E})}+Ca^{2}f^{(Ca^{2})}+\cdots \,,\end{eqnarray}$$

where $f^{(Ca)}$ , $f^{(CaRe_{E})}$ and $f^{(Ca^{2})}$ are the corrections in drop shape due to leading-order, $O(Re_{E})$ and $O(Ca)$ electric and viscous stresses which are to be determined later by invoking normal stress balance.

3.1 Drop shape

The $O(Ca)$ correction in drop shape is obtained as

(3.1) $$\begin{eqnarray}f^{(Ca)}=L_{2}^{(Ca)}P_{2}(\unicode[STIX]{x1D702}),\end{eqnarray}$$

where $P_{n}(\unicode[STIX]{x1D702})$ represents the $n$ th degree Legendre polynomial. The coefficient $L_{2}^{(Ca)}$ is obtained in the following form

(3.2) $$\begin{eqnarray}L_{2}^{(Ca)}=\frac{1}{8}\left(\frac{19\unicode[STIX]{x1D706}+16}{\unicode[STIX]{x1D706}+1}\right)+\frac{3}{4}\frac{M\unicode[STIX]{x1D6FA}_{T}}{(R+2)^{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}_{T}=R^{2}+1-2S+3(R-S)(3\unicode[STIX]{x1D706}+2)/5(\unicode[STIX]{x1D706}+1)$ is the Taylor discriminating function. The sole effect of the uniform electric field is to deform the drop into a prolate shape (represented by $\unicode[STIX]{x1D6FA}_{T}>0$ ) or an oblate shape (represented by $\unicode[STIX]{x1D6FA}_{T}<0$ ), while the sole effect of the extensional flow is always to deform the drop into a prolate shape. It is evident from (3.2) that the effect of the uniform electric field and uniaxial extensional flow on the drop shape combines linearly at this order.

The $O(CaRe_{E})$ correction in drop shape is obtained as

(3.3) $$\begin{eqnarray}f^{(CaRe_{E})}=L_{2}^{(CaRe_{E})}P_{2}(\unicode[STIX]{x1D702})+L_{4}^{(CaRe_{E})}P_{4}(\unicode[STIX]{x1D702}),\end{eqnarray}$$

where the coefficients $L_{2}^{(CaRe_{E})}$ and $L_{4}^{(CaRe_{E})}$ are obtained as

(3.4a,b ) $$\begin{eqnarray}L_{2}^{(CaRe_{E})}=M(R-S)\ell _{1},\quad L_{4}^{(CaRe_{E})}=M(R-S)\ell _{2},\end{eqnarray}$$

where $\ell _{1}$ and $\ell _{2}$ are known functions of $R$ , $S$ , $\unicode[STIX]{x1D706}$ and $M$ which we have given in appendix B.

The $O(Ca^{2})$ correction in drop shape is obtained as

(3.5) $$\begin{eqnarray}f^{(Ca^{2})}=L_{2}^{(Ca^{2})}P_{2}(\unicode[STIX]{x1D702})+L_{4}^{(Ca^{2})}P_{4}(\unicode[STIX]{x1D702}),\end{eqnarray}$$

where the coefficients $L_{2}^{(Ca^{2})}$ and $L_{4}^{(Ca^{2})}$ are obtained as

(3.6a,b ) $$\begin{eqnarray}L_{2}^{(Ca^{2})}=L_{2}^{(Ca)}(\ell _{3}+\ell _{4}L_{2}^{(Ca)}),\quad L_{4}^{(Ca^{2})}=L_{2}^{(Ca)}(\ell _{5}+\ell _{6}L_{2}^{(Ca)}),\end{eqnarray}$$

where $\ell _{3}{-}\ell _{6}$ are known functions of $R$ , $S$ , $\unicode[STIX]{x1D706}$ and $M$ which we have given in appendix B.

Now, substituting $f^{(Ca)}$ , $f^{(CaRe_{E})}$ and $f^{(Ca^{2})}$ in (2.12), we obtain the deformed drop shape as

(3.7) $$\begin{eqnarray}\displaystyle r_{s} & = & \displaystyle \left[1+CaL_{2}^{(Ca)}P_{2}(\unicode[STIX]{x1D702})+CaRe_{E}\left\{L_{2}^{(CaRe_{E})}P_{2}(\unicode[STIX]{x1D702})+L_{4}^{\left(CaRe_{E}\right)}P_{4}(\unicode[STIX]{x1D702})\right\}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left.Ca^{2}\left\{c_{v}+L_{2}^{(Ca^{2})}P_{2}\left(\unicode[STIX]{x1D702}\right)+L_{4}^{(Ca^{2})}P_{4}(\unicode[STIX]{x1D702})\right\}\right].\end{eqnarray}$$

The term $c_{v}=-(L_{2}^{(Ca)})^{2}/5$ is included at $O(Ca^{2})$ to conserve the drop volume. When the drop shape is nearly spherical, the degree of drop deformation is characterized by the deformation parameter of the form ${\mathcal{D}}=\left(L-B\right)/\left(L+B\right)$ , where $L=r_{s}\left(\unicode[STIX]{x1D703}=0\right)$ and $B=r_{s}\left(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2\right)$ are the semi-major and semi-minor axes of the deformed ellipsoidal drop shape. The deformation parameter ${\mathcal{D}}>0$ represents the prolate shape of the drop, while ${\mathcal{D}}<0$ represents the oblate shape of the drop. We obtain the expression for the deformation parameter which includes the combined effect of the uniform electric field and uniaxial extensional flow as

(3.8) $$\begin{eqnarray}\displaystyle {\mathcal{D}} & = & \displaystyle \displaystyle \frac{3L_{2}^{(Ca)}Ca}{4}+\left\{\frac{3L_{2}^{(CaRe_{E})}}{4}+\frac{5L_{4}^{(CaRe_{E})}}{16}\right\}CaRe_{E}\nonumber\\ \displaystyle & & \displaystyle +\displaystyle \,\left\{\frac{3L_{2}^{\left(Ca^{2}\right)}}{4}+\frac{5L_{4}^{(Ca^{2})}}{16}-\frac{3\left(L_{2}^{(Ca)}\right)^{2}}{16}\right\}Ca^{2}.\end{eqnarray}$$

3.2 Emulsion extensional rheology

Presence of a uniform electric field affects both the flow around the drop and drop shape. The altered flow field and drop shape can affect the rheological properties of a dilute suspension of Newtonian drops in the presence of a background uniaxial extensional flow. Here, we focus on the emulsion stress for a dilute emulsion with $\unicode[STIX]{x1D708}\ll 1$ , where $\unicode[STIX]{x1D708}$ is the drop volume fraction. The effective suspension stress $\unicode[STIX]{x1D6F4}$ for a dilute suspension of drops in linear flow is given as (Barthès-Biesel Reference Barthès-Biesel2012) $\unicode[STIX]{x1D6F4}=-p\boldsymbol{I}+2\unicode[STIX]{x1D71E}+\unicode[STIX]{x1D72E}^{(d)}$ , where $\unicode[STIX]{x1D72E}^{(d)}$ is the stresslet tensor. The stresslet tensor $\unicode[STIX]{x1D72E}^{(d)}$ can be represented in the following form (Barthès-Biesel Reference Barthès-Biesel2012)

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D72E}^{(d)}=\frac{3\unicode[STIX]{x1D708}}{4\unicode[STIX]{x03C0}}\int _{S_{d}}\left[\frac{1}{2}\left\{\left(\unicode[STIX]{x1D749}_{e}^{H}\boldsymbol{\cdot }\boldsymbol{n}\right)\boldsymbol{r}+\left(\left(\unicode[STIX]{x1D749}_{e}^{H}\boldsymbol{\cdot }\boldsymbol{n}\right)\boldsymbol{r}\right)^{\text{T}}-\frac{2}{3}\boldsymbol{I}\left(\unicode[STIX]{x1D749}_{e}^{H}\boldsymbol{\cdot }\boldsymbol{n}\right)\boldsymbol{\cdot }\boldsymbol{r}\right\}-\left(\boldsymbol{u}_{e}\boldsymbol{n}+\left(\boldsymbol{u}_{e}\boldsymbol{n}\right)^{\text{T}}\right)\right]\text{d}S_{d},\end{eqnarray}$$

where $S_{d}$ is the deformed drop surface. The stresslet tensor is generated due to the presence of suspended drops. Both charge convection and shape deformation affect the stresslet tensor. To quantify these effects, it is convenient for extensional flow to define a dimensional extensional viscosity $\unicode[STIX]{x1D707}_{ext}$ of the dilute suspension as $\unicode[STIX]{x1D707}_{ext}=(\bar{\unicode[STIX]{x1D6F4}}_{zz}-\bar{\unicode[STIX]{x1D6F4}}_{xx})/G=(\bar{\unicode[STIX]{x1D6F4}}_{zz}-\bar{\unicode[STIX]{x1D6F4}}_{yy})/G$ , where the dimensional stress component $\bar{\unicode[STIX]{x1D6F4}}_{ij}$ can be expressed as $\bar{\unicode[STIX]{x1D6F4}}_{ij}=\unicode[STIX]{x1D707}_{e}G\unicode[STIX]{x1D6F4}_{ij}$ . Now, substituting different terms we obtain the extensional viscosity ratio (also termed Trouton’s ratio which is defined as the ratio between extensional viscosity and shear viscosity) as $\unicode[STIX]{x1D702}_{ext}=\unicode[STIX]{x1D707}_{ext}/\unicode[STIX]{x1D707}_{e}=\unicode[STIX]{x1D702}_{ext}^{(0)}+Re_{E}\unicode[STIX]{x1D702}_{ext}^{\left(Re_{E}\right)}+Ca\unicode[STIX]{x1D702}_{ext}^{(Ca)}$ , where $\unicode[STIX]{x1D702}_{ext}^{\left(0\right)}$ represents the extensional viscosity ratio when the charge convection and shape deformation effects are absent, $\unicode[STIX]{x1D702}_{ext}^{\left(Re_{E}\right)}$ and $\unicode[STIX]{x1D702}_{ext}^{(Ca)}$ represent the explicit effect of charge convection and shape deformation on the extensional viscosity ratio. Expressions for $\unicode[STIX]{x1D702}_{ext}^{(0)}$ , $\unicode[STIX]{x1D702}_{ext}^{\left(Re_{E}\right)}$ and $\unicode[STIX]{x1D702}_{ext}^{(Ca)}$ are obtained as

(3.10) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D702}_{ext}^{(0)}=3+\unicode[STIX]{x1D708}\left[\frac{3}{2}\frac{\left(5\unicode[STIX]{x1D706}+2\right)}{\left(\unicode[STIX]{x1D706}+1\right)}+\frac{27}{5}\frac{M\left(S-R\right)}{\left(R+2\right)^{2}\left(\unicode[STIX]{x1D706}+1\right)}\right],\\[14.0pt] \displaystyle \unicode[STIX]{x1D702}_{ext}^{\left(Re_{E}\right)}=\unicode[STIX]{x1D708}\left[\frac{81M\left(R-S\right)\left\{5\left(R+2\right)^{2}+6M(R-S)\right\}\left(69R^{2}+130R-54SR-80S+40\right)}{875\left(\unicode[STIX]{x1D706}+1\right)^{2}\left(3R+4\right)\left(R+2\right)^{5}}\right],\\[21.0pt] \displaystyle \unicode[STIX]{x1D702}_{ext}^{(Ca)}=\unicode[STIX]{x1D708}\left[L_{2}^{(Ca)}\left(\frac{9}{70}\frac{41\unicode[STIX]{x1D706}+4+25\unicode[STIX]{x1D706}^{2}}{\left(1+\unicode[STIX]{x1D706}\right)^{2}}-\frac{162}{175}\frac{M(R-S)\left(13\unicode[STIX]{x1D706}R+15R-12-16\unicode[STIX]{x1D706}\right)}{\left(1+\unicode[STIX]{x1D706}\right)^{2}\left(R+2\right)^{3}}\right)\right].\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

5.1 Validation

Before investigating the combined influence of uniaxial extensional flow and uniform electric field, we compare our theoretical and numerical results with the existing results for the following two limiting cases: (i) drop deformation in a uniaxial extensional flow, and (ii) drop deformation in a uniform electric field. When the electric field is absent, the present analytical result for drop deformation recovers the analytical result of Barthés-Biesel & Acrivos (Reference Barthés-Biesel and Acrivos1973).

Figure 3(a) depicts the variation of drop deformation with capillary number in uniaxial extensional flow. It is evident that the present analytical result compares very well with the numerical result of Stone & Leal (Reference Stone and Leal1989) for small values of Ca. With increase in Ca, the analytical result underpredicts the deformation. However, our numerical solution compares very well even in the large deformation regime. When the extensional flow is absent, the present analytical result for drop deformation recovers the analytical result of Ajayi (Reference Ajayi1978) and Feng (Reference Feng1999). Figure 3(b) depicts the variation of drop deformation with electric capillary number ( $Ca_{E}=\unicode[STIX]{x1D700}_{e}E_{\infty }^{2}a/\unicode[STIX]{x1D6FE}$ ) in a uniform electric field. Note that $Ca_{E}$ represents the strength of electric stress (which tries to deform the drop) relative to the capillary stress (which resists the drop deformation). Similar to the drop deformation in uniaxial extensional flow, the present analytical result compares very well with the experimental result of Ha & Yang (Reference Ha and Yang2000a ) for small values of $Ca_{E}$ . However, the agreement between our numerical result and the experiment of Ha & Yang (Reference Ha and Yang2000a ) is over the whole $Ca_{E}$ regime under consideration.

Figure 3.

(a) Variation of droplet deformation with capillary number in uniaxial extensional flow for $\unicode[STIX]{x1D706}=1$ . (b) Variation of droplet deformation with electric capillary number in uniform electric field for $R=10$ , $S=1.37$ , $\unicode[STIX]{x1D706}=0.874$ and $Re_{E}=0.01$ . We consider $Re=0.01$ for all the simulations to simulate low Reynolds number flow.

5.2 Combined effect of uniform electric field and uniaxial extensional flow on the drop deformation

5.2.1 Asymptotic theory

Figure 4(a) shows the variation of drop deformation with Mason number for a perfectly dielectric drop which is suspended in another perfectly dielectric medium (PD–PD system). Although the analytical solution is obtained for a leaky dielectric drop suspended in a leaky dielectric medium (LD–LD system), substitution of $R=S$ into (3.8) yields the drop deformation for the PD–PD system. Note that $M=0$ represents the absence of electric field (i.e. only extensional flow), while $M>0$ represents the combined presence of electric field and extensional flow. Figure 4(a) shows that drop deformation increases with increase in $M$ (which represents increasing strength of electric field as compared to strain rate) for both values of the permittivity ratio. This increase in drop deformation with increase in Mason number can be explained by the fact that the independent effect of extensional flow and uniform electric field is to deform the drop into a prolate shape. Thus, the combined effect leads to larger deformation.

Figure 4.

(a) Variation of drop deformation with Mason number for $Ca=0.1$ and $\unicode[STIX]{x1D706}=0.1$ . (b) Variation of drop deformation with permittivity ratio for $Ca=0.1$ and $\unicode[STIX]{x1D706}=0.1$ .

Figure 5.

(a) Schematic representation of streamline in and around the drop in extensional flow. Variation of the jump in electric traction on the drop surface for (b) $S=5$ and (c) $S=0.2$ .

To investigate the effect of permittivity ratio on the drop deformation, we plot the variation of drop deformation with the permittivity ratio for the PD–PD system in figure 4(b). We have also plotted the electric-field-induced drop deformation which is the $M=\infty$ limit (i.e. absence of imposed extensional flow). Figure 4(b) shows that drop always deforms to a prolate shape (irrespective of the value of $S$ ) in the combined presence of an electric field and extensional flow. Figure 4(b) also shows that in the combined presence of a uniform electric field and extensional flow, the drop deformation is always larger as compared with either electric-field-induced or extensional-flow-induced drop deformation.

The physical reason for the prolate deformation of a PD drop and associated variation with permittivity ratio in the combined presence of electric field and extensional flow can be understood if we look into the independent effects of extensional flow and uniform electric field. The extensional-flow-induced drop deformation is due to the jump in the normal component of hydrodynamic traction. Figure 5(a) depicts a schematic drawing of the streamline pattern due to the imposed extensional flow. The flow at the drop surface is directed from the equator ( $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ ) towards the poles ( $\unicode[STIX]{x1D703}=0,\unicode[STIX]{x03C0}$ ) and this kind of flow is always associated with a jump in hydrodynamic traction which stretches the drop along the extension axis (i.e. prolate shape). On the other hand, the electric-field-induced deformation is due to the jump in normal component of electric traction. The distribution of the jump in the normal component of electric traction at the drop surface is shown in figure 5(b) for $S=5$ and figure 5(c) for $S=0.2$ . The jump in normal electric traction for $S=5$ is radially outward and largest near the poles ( $\unicode[STIX]{x1D703}=0,\unicode[STIX]{x03C0}$ ), while the magnitude reduces near the equator ( $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ ). Thus for $S=5$ the drop is stretched near the poles (the same is true for $S>1$ ). On the other hand, the jump in electric traction for $S=0.2$ is radially inward and largest near the equator ( $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ ), while the magnitude reduces near the poles ( $\unicode[STIX]{x1D703}=0,\unicode[STIX]{x03C0}$ ). Thus for $S=0.2$ the drop is compressed near the equator (the same is true for $S<1$ ). In both cases the drop is deformed into a prolate shape. The permittivity ratio has a large impact on the electric-field-induced drop deformation (as depicted in figure 4 b) due to the dependence of electric traction on the permittivity ratio. Permittivity ratio significantly modulates the field lines near the drop surface. For example, when $S>1$ (or $S<1$ ) the density of field lines inside the drop is smaller (or larger) than outside. For the particular case of $S=1$ , the field lines are not at all distorted by the drop and the jump in electric traction vanishes. Thus for $S=1$ the electric field has no effect on the drop deformation. So, the independent effect of the extensional flow and electric field is always to deform the drop into a prolate shape. For given values of Ca and $\unicode[STIX]{x1D706}$ , there is a critical permittivity ratio beyond which the electric-field-induced drop deformation is larger than the extensional-flow-induced drop deformation. This is due to the larger magnitude of the jump in the normal component of electric traction as compared with the jump in the normal component of hydrodynamic traction. In the combined presence of a uniform electric field and extensional flow, the drop deformation is always larger as compared with either electric-field-induced or extensional-flow-induced drop deformation. However, for $S=1$ the jump in electric traction vanishes and only the extensional flow deforms the drop. This explains the fact that the drop deformation with $M=2$ and $S=1$ is exactly equal to the drop deformation solely due to extensional flow as depicted in figure 4(b).

Figure 6.

(a) Variation of drop deformation with Mason number for $Ca=Re_{E}=\unicode[STIX]{x1D706}=0.1$ . (b) Variation of drop deformation with conductivity ratio for $Ca=Re_{E}=\unicode[STIX]{x1D706}=0.1$ and $M=1$ .

The deformation behaviour of LD–LD system is depicted in figure 6(a) where we have shown the variation of drop deformation with the Mason number. Figure 6(a) shows that the drop deformation increases with the Mason number for $(R,S)=(2.0,0.5)$ . This is due to the positive value of the Taylor discriminating function ( $\unicode[STIX]{x1D6FA}_{T}>0$ ), which means the electric field favours prolate deformation. Thus, similar to the PD–PD system, both the extensional flow and electric field deform the drop into a prolate shape for $\unicode[STIX]{x1D6FA}_{T}>0$ . However, in sharp contrast to the PD–PD system, the drop deformation decreases with the Mason number for $(R,S)=(0.5,2.0)$ . This is due to the negative value of the Taylor discriminating function ( $\unicode[STIX]{x1D6FA}_{T}<0$ ) which means the electric field favours oblate deformation. So, for $\unicode[STIX]{x1D6FA}_{T}<0$ the electric-field-induced and extensional-flow-induced drop deformations are opposite to each other. For smaller values of $M$ , extensional flow dominates over the electric field and the drop deforms to a prolate shape as depicted in figure 6(a). However, for relatively larger values of $M$ , the electric field dominates over the extensional flow and the drop may deform into an oblate shape. To investigate the effect of electrical parameters, we plot the variation of drop deformation with the conductivity ratio ( $R$ ) for different values of permittivity ratio ( $S$ ) in figure 6(b). The first important thing to note from figure 6(b) is that, depending on the values of $R$ and $S$ , the drop can deform into a prolate or oblate shape in the combined presence of an electric field and extensional flow. For $R\gg 1$ the drop deformation is independent of $S$ . This is due to the fact that in the limit $R\rightarrow \infty$ , the drop behaves as a perfectly conducting drop. For a perfectly conducting drop, the drop surface becomes an equipotential surface. The electric field inside the drop vanishes which means the electric permittivity of the drop fluid has no impact on the distribution of electric potential and electric traction. Electric permittivity has a noteworthy impact on the drop deformation when $R\ll 1$ as depicted in figure 6(b).

Figure 7.

Variation of drop deformation with Mason number for (a) $(R,S)=(2.0,0.5)$ and (b) $(R,S)=(0.1,2.0)$ . Other parameters are taken as $\unicode[STIX]{x1D706}=0.1$ and $Ca=0.1$ .

To study the pivotal role of surface charge convection in the drop deformation, we show the drop deformation in the presence of surface charge convection (i.e. $Re_{E}>0$ ) and compare it with the case of no surface charge convection (i.e. $Re_{E}=0$ ). Previous studies on drop deformation in a uniform electric field established the fact that surface charge convection has a strong influence on the drop deformation (Lanauze et al. Reference Lanauze, Walker and Khair2015; Das & Saintillan Reference Das and Saintillan2017). In the combined presence of a uniform electric field and extensional flow, we first plot the variation of drop deformation with Mason number considering $(R,S)=(2,0.5)$ in figure 7(a). Figure 7(a) shows that the charge convection increases the drop deformation. Surface charge convection leads to a greater increase in drop deformation for larger values of the Mason number. Figure 7(b) shows that the charge convection can decrease or increase the drop deformation depending on the value of Mason number for $(R,S)=(0.1,2.0)$ . A closer look into figure 7(b) shows that when the drop deforms into prolate shape, the charge convection can either decrease (for $M=1$ ) or increase (for $M=2$ ) the drop deformation. But when the drop deforms into an oblate shape, the charge convection always decreases the drop deformation (e.g. $M=3$ ).

Figure 8.

Schematic representation of two different types of EHD flow in and around drop in the presence of a uniform electric field for (a) $R>S$ and (b) $R<S$ .

Previous studies on the drop deformation in a uniform electric field have established the fact that charge convection enhances prolate deformation, while it acts to reduce the oblate deformation (Feng Reference Feng1999). However, in the combined presence of a uniform electric field and uniaxial extensional flow, we obtain either an increase or decrease in drop deformation due to charge convection when the drop is deformed into a prolate shape, while an oblate drop is found to deform less in the presence of charge convection. To investigate the physical reason for the charge-convection-induced increase/decrease in drop deformation, we consider two different cases: (i) $R>S$ (e.g. $R=2$ and $S=0.5$ ), (ii) $R<S$ (e.g. $R=0.1$ and $S=2$ ). To understand the charge-convection-induced drop deformation, we look into the effect of fluid flow on the distribution of surface charges. In the absence of convection, the surface charge distribution is governed by the ohmic conduction (Salipante & Vlahovska Reference Salipante and Vlahovska2010). The distribution of charges at the drop surface is governed by the ratio $R/S=t_{e}/t_{i}$ , where $t_{e}=(\unicode[STIX]{x1D700}_{e}/\unicode[STIX]{x1D70E}_{e})$ and $t_{i}=(\unicode[STIX]{x1D700}_{i}/\unicode[STIX]{x1D70E}_{i})$ are the charge relaxation time scales of suspending phase and drop phase, respectively. Charge relaxation time scale physically signifies the time required for charges to reach a steady configuration due to ohmic conduction (Melcher & Taylor Reference Melcher and Taylor1969). For $R>S$ or equivalently $t_{i}<t_{e}$ the ohmic conduction in the drop phase is much faster (due to small time scale) than the ohmic conduction in the suspending phase. Consequence of this is the charging of the drop surface by the bulk charges coming from the drop phase. In the presence of a uniform electric field acting along the positive $z$ direction, the fast conduction in the drop phase leads to accumulation of positive charges at the upper hemispherical surface ( $0\leqslant \unicode[STIX]{x1D703}<\unicode[STIX]{x03C0}/2$ ), while negative charges accumulate at the lower hemispherical surface ( $\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D703}\leqslant \unicode[STIX]{x03C0}$ ). As the charges at each pole encounter the opposite charges of the nearby electrode (e.g. the upper hemisphere encounters the negative electrode), the charges at each pole will be attracted by the nearby electrode. This attraction causes a jump in tangential electric traction which drives an EHD flow from the equator towards the poles (refer to figure 8 a). The physical situation is reversed for $R<S$ as depicted in figure 8(b). For $R<S$ or equivalently $t_{e}<t_{i}$ the ohmic conduction in the suspending phase is much faster (due to the small time scale) than the ohmic conduction in drop phase. Consequence of this is the charging of the drop surface by the bulk charges coming from the suspending phase. In the presence of a uniform electric field acting along the positive $z$ direction, the fast conduction in the suspending phase leads to accumulation of negative charges at the upper hemispherical surface ( $0\leqslant \unicode[STIX]{x1D703}<\unicode[STIX]{x03C0}/2$ ), while positive charges accumulate at the lower hemispherical surface ( $\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D703}\leqslant \unicode[STIX]{x03C0}$ ). As the charges at each pole encounter similar charges of the nearby electrode, the charges at each pole will be repelled by the nearby electrode. This repulsion causes a jump in tangential electric traction which drives an EHD flow from the poles towards the equator (refer to figure 8 b).

Figure 9.

(a) Variation of surface charge distribution over the drop interface for $R=2$ , $S=0.5$ , $\unicode[STIX]{x1D706}=0.1$ and $M=3$ . (b) Variation of surface charge distribution over the drop interface for $R=0.1$ , $S=2$ , $\unicode[STIX]{x1D706}=0.1$ and $M=1$ .

The previously mentioned distribution of surface charges is likely to be disturbed by the extensional and EHD components of fluid flow. The combined influence of the extensional flow and the EHD flow on the charge distribution is governed by $R$ , $S$ , $\unicode[STIX]{x1D706}$ and $M$ . In figure 9 we show the distribution of surface charges $q_{s}=q_{s}^{(0)}+Re_{E}q_{s}^{\left(Re_{E}\right)}$ , where $q_{s}^{(0)}$ and $q_{s}^{\left(Re_{E}\right)}$ are the leading-order and $O(Re_{E})$ charge distributions, respectively. When $R>S$ the velocity at the drop surface always runs from the equator to the poles as both the extensional flow and EHD flow act in the same direction. In the presence of surface charge convection (i.e. $Re_{E}>0$ ), this equator-to-poles surface velocity drives the charges from the equator to the poles which leads to an increase in surface charge density near the poles and decrease in charge density near the equator, as depicted in figure 9(a). As the charges at each pole are attracted by the nearby electrode, the increase in surface charge density near the poles drastically increases the jump in the normal component of electric traction near the poles which finally results in larger drop deformation for $Re_{E}>0$ . The increase in the jump of the normal component of electric traction eventually increases the drop deformation because $\unicode[STIX]{x1D6FA}_{T}>0$ which means the electric-field-induced deformation (prolate shape) augments the extensional-flow-induced deformation (prolate shape). Thus, for $R>S$ the charge convection has a destabilizing effect on the drop shape. Now, we discuss the case of $R<S$ for which the velocity at the drop surface can run in either direction (equator to poles or poles to equator) as the extensional flow and EHD flow act opposite to each other. For $M=1$ , the extensional flow dominates and the surface velocity runs from the equator to poles. Thus, in the presence of surface charge convection (i.e. $Re_{E}>0$ ), this equator-to-poles surface velocity drives the charges from the equator to the poles which leads to an increase in surface charge density near poles and a decrease in charge density near the equator as depicted in figure 9(b). As the charges at each pole are repelled by the nearby electrode, the increase in surface charge density near the poles drastically increases the jump in the normal component of electric traction near the poles which finally results in smaller drop deformation for $Re_{E}>0$ . The increase in the jump of the normal component of electric traction eventually reduces the drop deformation because $\unicode[STIX]{x1D6FA}_{T}<0$ (for $(R,S)=(0.1,2)$ ) which means the electric-field-induced deformation (oblate shape) opposes the extensional-flow-induced deformation (prolate shape). On the contrary, for $M=2$ the charge convection effect increases the drop deformation (as depicted in figure 7 b). This is due to the fact that for $M=2$ the EHD flow dominates over the extensional flow and the surface velocity is directed from the poles to the equator. This poles-to-equator flow eventually drives the charges away from the poles and reduces the charge density near the poles. Reduction in surface charge density yields a significant decrease in the jump in the normal component of electric traction which finally results in larger drop deformation for $Re_{E}>0$ . Thus, for $R<S$ the charge convection has a stabilizing/destabilizing effect on the drop shape.

Figure 10.

(a) Variation of drop deformation with capillary number for $S=5$ , $\unicode[STIX]{x1D706}=1$ and $M=2$ . (b) Variation of drop deformation with capillary number for $R=0.5$ , $S=2$ , $\unicode[STIX]{x1D706}=1$ , $Re_{E}=0.1$ and $M=2$ . Drop shapes (line represent higher-order analytical theory and marker represents numerical) are shown as insets.

5.2.2 Comparison of analytical solution against numerical simulation

To verify the applicability of the present asymptotic theory in the combined presence of electric field and extensional flow, we compare the analytical results with the results obtained from full-scale numerical simulations. We plot variation of drop deformation with capillary number in figure 10(a) for the PD–PD system with $S=5$ , and figure 10(b) for the LD–LD system with $(R,S)=(0.5,2)$ . It is evident from these panels that both the leading-order linear theory (i.e. taking only the $O(Ca)$ deformation into account) and the higher-order nonlinear theory (i.e. taking $O(Ca)$ , $O(Ca^{2})$ and $O(CaRe_{E})$ deformations into account) compare very well with the numerical solution for small values of Ca. With an increase in Ca, the higher-order nonlinear theory gives a better match with the numerical solution as compared with the leading-order linear theory. Several drop shapes are also shown as insets in figure 10. Analytically obtained drop shapes compare very well with the numerically obtained drop shapes in the small deformation regime. In the large deformation regime, the numerical solution shows more elongated drop shapes which deviate from the analytical solution. Thus the higher-order nonlinear theory under predicts the deformation as we go for larger values of Ca. Important thing to note here is that for the EHD property ratios under consideration, the combination of $O(Ca^{2})$ and $O(CaRe_{E})$ corrections leads to an increase in deformation which indicates that with an increase in Ca the drop will break eventually. So, higher-order terms have a destabilizing effect on the drop shape.

Figure 11.

(a) Variation of drop deformation with Mason number. (b) Variation of semi-major and semi-minor axes of the deformed drop with Mason number for $R=2$ and $S=0.5$ . Other parameters are taken as for $\unicode[STIX]{x1D706}=1$ , $Ca=0.05$ and $Re_{E}=0.1$ .

To investigate the drop deformation beyond the small deformation limit, we plot the variation of drop deformation over a wide range of Mason numbers in figure 11(a). As the asymptotic theory is valid for $M\sim 1$ , in figure 11(a) we show only numerical results. Also note that the magnitude of the deformation obtained from the numerical simulations (as depicted in figure 11 a) is beyond the regime of validity of asymptotic theory. Figure 11(a) shows that for $(R,S)=(2,0.5)$ the drop deforms into a prolate shape for all values of $M$ . This is due to the fact that the Taylor discriminating function $\unicode[STIX]{x1D6FA}_{T}>0$ which means the uniform electric field aids the drop deformation due to uniaxial extensional flow. Drop deformation significantly increases for larger values of $M$ . On the other hand, for $(R,S)=(0.5,2)$ we obtain prolate deformation of the drop when $M$ is small. But with increase in $M$ , the magnitude of drop deformation decreases and eventually the drop deforms into an oblate shape. This prolate-to-oblate shape transition is due to the fact that the Taylor discriminating function $\unicode[STIX]{x1D6FA}_{T}<0$ for $(R,S)=(0.5,2)$ which means the uniform electric field counteracts the extensional-flow-induced drop deformation. For relatively smaller values of $M$ , uniaxial extensional flow dominates and the drop deforms to a prolate shape, while for larger values of $M$ the uniform electric field dominates and deforms the drop into an oblate shape. For this case, there is a critical value of $M$ for which the deformation is zero which signifies the fact the normal components of hydrodynamic and electric tractions cancel each other. Note that the deformation parameter ${\mathcal{D}}$ lumps the two semi-axes in one single parameter. To show the influence of electric field on the individual semi-axes of the drop, we plot the variation of semi-major axis ( $L$ ) and semi-minor axis ( $B$ ) with the Mason number in figure 11(b) for $(R,S)=(2,0.5)$ . Note that in the present study two semi-axes are sufficient to describe the drop shape completely because the drop shape is axisymmetric. Figure 11(b) shows that with an increase in Mason number the semi-major axis becomes more elongated, while the semi-minor axis becomes more compressed. This explains the rapid increase in ${\mathcal{D}}$ with Mason number shown in figure 11(a). The opposite behaviour of the two semi-axes is due to the fact that the drop volume is conserved. Thus, the decrease in length of the semi-minor axis is always associated with the increase in length of the semi-major axis.

Figure 12.

(a) Variation of the normalized leading-order extensional viscosity ratio with viscosity ratio. (b) Variation of the normalized leading-order extensional viscosity ratio with strain rate ( $M^{-1}$ ) for $\unicode[STIX]{x1D706}=0.1$ .

5.3 Effect of uniform electric field on the emulsion extensional viscosity

The effective extensional viscosity represents the bulk behaviour of an emulsion in response to a tensile or elongation stress. For a Newtonian fluid the extensional viscosity is three times the shear viscosity (Shroff et al. Reference Shroff, Cancio and Shida1977). However, this relation does not hold for emulsions. Drops perturb the imposed extensional flow and thereby induce the stresslet tensor and alter the effective extensional viscosity of the emulsion. The electric field affects the extensional viscosity in two different ways: (i) the EHD flow interacts with the extensional flow, and (ii) the electric field deforms the drop. Towards investigating the effect of electric field, we first consider the case in which the drops remain spherical and charge convection is absent. We plot the variation of $(\unicode[STIX]{x1D702}_{ext}^{(0)}-3)/\unicode[STIX]{x1D708}$ in figure 12(a) for two different cases: (i) $R>S$ (e.g. $R=2$ and $S=0.5$ ), and (ii) $R<S$ (e.g. $R=0.5$ and $S=2$ ). In the absence of an electric field (i.e. $M=0$ ), $\unicode[STIX]{x1D702}_{ext}^{(0)}$ increases with viscosity ratio. This increase in extensional viscosity ratio is attributed to the fact that a highly viscous drop severely decreases the magnitude of the velocity in and around the drop, which further increases the rate of dissipation in the flow. Figure 12(a) shows that application of a uniform electric field significantly decreases (or increases) $\unicode[STIX]{x1D702}_{ext}^{(0)}$ when $R>S$ (or $R<S$ ). The physical reason for this increase (or decrease) in $\unicode[STIX]{x1D702}_{ext}^{(0)}$ can be found by looking into the nature of the flow field. For $R>S$ , the EHD flow runs from equator to poles, which aids the extensional flow and thereby decreases $\unicode[STIX]{x1D702}_{ext}^{(0)}$ . On the other hand, for $R<S$ the EHD flow runs from poles to equator, which resists the extensional flow and thereby increases $\unicode[STIX]{x1D702}_{ext}^{(0)}$ . Another important thing to note from figure 12(a) is that the effect of electric field on $\unicode[STIX]{x1D702}_{ext}^{(0)}$ diminishes for larger values of $\unicode[STIX]{x1D706}$ . This is attributed to the fact that the EHD flow ceases to exist for larger values of $\unicode[STIX]{x1D706}$ . Another important thing to note from figure 12(a) is that a higher value of $\unicode[STIX]{x1D706}$ always leads to a higher value of $\unicode[STIX]{x1D702}_{ext}^{(0)}$ for $R>S$ . This is due to fact that that the velocity at the drop interface runs from equator to poles irrespective of the value of $M$ . However, for $R<S$ we obtain (i) larger $\unicode[STIX]{x1D702}_{ext}^{(0)}$ for larger value of $\unicode[STIX]{x1D706}$ when $M=1$ , and (ii) larger $\unicode[STIX]{x1D702}_{ext}^{(0)}$ for lower value of $\unicode[STIX]{x1D706}$ when $M=3$ . This is attributed to the fact that for $M=1$ the surface velocity is dominated by the imposed extensional flow which yields equator-to-poles flow at the drop interface. On the other hand, for $M=3$ the poles-to-equator EHD flow dominates over the extensional flow at the drop interface. This makes larger $\unicode[STIX]{x1D702}_{ext}^{(0)}$ for a lower value of $\unicode[STIX]{x1D706}$ when $M=3$ .

A very interesting point to note from the leading-order extensional viscosity is the strain-rate dependency of the effective viscosity. When the electric field is absent (i.e. $M=0$ ), it is evident from (3.10) that $\unicode[STIX]{x1D702}_{ext}^{(0)}$ is solely a function of viscosity ratio. But when the electric field is present (i.e. $M>0$ ), $\unicode[STIX]{x1D702}_{ext}^{(0)}$ becomes strain-rate dependent because the Mason number ( $M=\unicode[STIX]{x1D700}_{e}E_{\infty }^{2}/\unicode[STIX]{x1D707}_{e}G$ ) is inversely proportional to the strain rate. To show the strain-rate-dependent property of the emulsion, we plot the variation of $\unicode[STIX]{x1D702}_{ext}^{(0)}$ with the normalized strain rate (i.e. $M^{-1}$ ) in figure 12(b). For $R>S$ (e.g. $R=2$ and $S=0.5$ ) the extensional viscosity increases with increase in strain rate, while for $R<S$ (e.g. $R=0.5$ and $S=2$ ) the extensional viscosity decreases with increase in strain rate. Thus, the emulsion exhibits strain-rate thickening behaviour for $R>S$ . The strain-rate thickening behaviour for $R>S$ is attributed to the fact that with an increase in strain rate (or decrease in Mason number) the magnitude of the EHD flow reduces. As the EHD flow aids the extensional flow for $R>S$ , a decrease in EHD flow effectively increases the resistance to flow and thereby increases the effective viscosity. On the other hand, the emulsion exhibits strain-rate thinning behaviour for $R<S$ . The strain-rate thinning behaviour for $R<S$ is attributed to the fact that with an increase in strain rate (or decrease in Mason number) the magnitude of the EHD flow reduces. As the EHD flow retards the extensional flow for $R<S$ , a decrease in EHD flow effectively decreases the resistance to flow and thereby decreases the effective viscosity.

For the PD–PD system (i.e. when $R=S$ ) the external electric field has no effect on the leading-order effective extensional viscosity ratio. This is due to the fact that for a PD–PD system the EHD flow vanishes. However, shape deformation can affect the effective extensional viscosity at $O(Ca)$ even for the PD–PD system. Shape deformation modifies the distribution of the electric potential and flow field at $O(Ca)$ and thereby perturbs the imposed extensional flow. To investigate the effect of shape deformation on the emulsion rheology, we plot the variation of $(\unicode[STIX]{x1D702}_{ext}-3)/\unicode[STIX]{x1D708}$ with viscosity ratio for $R=S=5$ in figure 13(a). Shape deformation is found to increase the effective extensional viscosity as evident from figure 13(a). This increase in $\unicode[STIX]{x1D702}_{ext}$ is attributed to the prolate shape of the drop. As the drops are stretched along the extension axis, the interfacial tension combined with the large curvature near the poles resists any further extension of the fluid element along the extension axis. Thus, a prolate shape is associated with a larger extensional viscosity as compared to a spherical shape. Things become more interesting for the LD–LD system due to the combined presence of shape deformation and charge convection. To investigate the effect of shape deformation and surface charge convection, we plot the variation of $(\unicode[STIX]{x1D702}_{ext}-3)/\unicode[STIX]{x1D708}$ with viscosity ratio for $(R,S)=(0.1,2)$ in figure 13(b). First we consider the case of a deformable drop with no charge convection (i.e. $Ca=0.1$ and $Re_{E}=0$ ). In this case the shape deformation can increase (or decrease) the effective extensional viscosity for smaller (or larger) values of viscosity ratio. Note that in this case the drop deforms into an oblate shape (considering $M=3$ ). As the drops are stretched perpendicular to the extension axis, the interfacial tension combined with large curvature near the equator augments any further extension of the fluid element along the extension axis. Thus, an oblate shape is associated with smaller effective extensional viscosity as compared to a spherical shape. But for a LD drop the shape deformation also modifies the EHD flow around the drop, which resists the extensional flow for $R<S$ (this is the case under consideration). Thus, the mode of deformation leads to a decrease in effective viscosity, while the EHD flow acts opposite to that of the extensional flow. Therefore, depending on the relative strength of the electric field and extensional flow, either effect can dominate and the extensional viscosity can increase or decrease. In the large viscosity ratio limit, the EHD flow vanishes and the effective viscosity reduces due to the oblate shape. Now, we consider the case of a spherical drop with charge convection (i.e. $Ca=0$ and $Re_{E}=0.1$ ). Charge convection alters the surface charge distribution and creates $O$ ( $Re_{E}$ ) electric traction. This electric traction induces a flow which perturbs the imposed extensional flow and thereby modifies the extensional viscosity. In this case charge convection modifies the EHD flow and reduces the effective extensional viscosity. Effect of charge convection vanishes for larger viscosity ratio as the surface velocity vanishes in the limit $\unicode[STIX]{x1D706}\rightarrow \infty$ . The combined effect of shape deformation and charge convection can be obtained by linearly combining these two effects which leads to an increase or decrease the effective extensional viscosity depending on the relative magnitude of the electrohydrodynamic parameters.

Figure 13.

(a) Variation of the normalized extensional viscosity ratio with viscosity ratio for $R=S=5$ and $M=1$ . (b) Variation of the normalized extensional viscosity ratio with viscosity ratio for $R=0.1$ , $S=2$ and $M=3$ .